First, notice that because \(C\)  is supposed to be finite dimensional, its
realization is geodesic by Proposition REF . The equivalence
between (i), (ii) and (iii) is due to Gromov [1]}, see
also Thm. 5.20212. The equivalence between (iii) and (iv) is
immediate and the equivalence between (iv) and (v) was shown in
Theorem REF .
The equivalence between (i'), (ii') and (iii') is due to
Gromov Thm. 5.5207. The equivalence between (iii') and (iv')
follows from the equivalence between (i) and (v).
